Simultaneous Equations: Notes on Estimation
You will be responsible for three methods of estimation:
OLS
ILS (Indirect least squares) this method is appropriate for exactly identified equations.
1. Estimate the reduced form
2. Solve for the structural coefficients from the reduced form coefficients
2SLS (Two Stage Least Squares) This method is appropriate for exactly or over identified
equations. If the equation is exactly identified 2SLS and ILS give identical results.
1. Regress each of the current endogenous variables which appear as explanatory
variable in the equation you are estimating on all the predetermined variables in
the system.
2. Estimate the equation substituting the fitted values from step 1 for the current
endogenous variables in the equation.
Example
Where y,x,z are current endogenous variables and m,g,n,p are exogenous variables. Suppose that
we want to estimate the first equation, the y equation. In the first stage we would regress both x
and z on m, g, n, and p. In the second stage we would regress y on the fitted values of z and x
from the first stage. (You should verify that each equation is over identified.)
Stata does this for you using the instrumental variables command.
reg y x z p (m g n p)
Hausman Test: If x and z are not correlated with e1 in the above example then the y equation can be
estimated using OLS. 2SLS which is a form of Instrumental Variables Estimation involves a
loss of efficiency and should not be used if it is not necessary. The following variant of the
Hausman test is used to test for this possibility.
1. Estimate the first stage regression for x and z. Take the residuals from these estimations.
Call these residuals .
2. Estimate the following and test the
hypothesis . If we fail to reject this hypothesis we conclude that the errors
and the explanatory variables are not correlated and that OLS is appropriate.
